The class of simply connected sets, which is the object of study of the present paper, is closely related to the class of unicoherent sets introduced by VietorisJ and Kuratowski. § A connected set is unicoherent if, however it be expressed as the sum of two connected and relatively closed subsets, the common part of the latter is connected. For locally connected metric sets the two classes coincide. In order that a connected and locally arcwise connected subset M of the plane be simply

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... , it is necessary and sufficient that the interior of every simple closed curve lying in M be a subset of M. The notion of simple connectedness in the weak sense is also defined. The properties of sets of these types have a variety of applications and furnish an interesting background for a number of well known theorems.